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Full text of "Treatise On Analysis Vol-Ii"

50       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

n~1(z)9 and let K(x) be a set with the properties enunciated above. Then
V(z)  (n(K(x)J)0 is an open neighborhood of z, and therefore n~l(y(z)) is an
open neighborhood of n~l(z) n An. Since An is compact, there are a finite
number of points zi en(An) such that the sets n~1(V(zi)) (1 ^i<>pn, say)
cover An . Let Kin be the set K(x) corresponding to V(zf), so that the interiors
of the sets n(Kin) form an open covering of 7r(AJ in E/G. Then it is clear that
the Kin (n ^ 1, 1 g i ^ pn for each ) satisfy the required conditions.

This being so, the restriction of n to Kn is a homeomorphism of Kn onto
the subspace n(Kn) of E/G, because E/G is Hausdorff (12.3.6), and this sub-
space is therefore compact and metrizable. The result therefore follows
from (12.4.7).

(12.10.11) Let G (resp. G') be a topological group acting continuously on a
topological space E (resp. E'). Then G x G' acts continuously on E x E' by
(s, s') - (x, x') = (s * x, s' - x'), and the mapping CD defined by

(Q x GO - (x, *')^((G  x\ (G' - *'))
is a homeomorphism of(E x E')/(G x G') onto (E/G) x (E'/G').

It is immediately checked that co is bijective, and it is continuous by virtue
of (12.10.5). Moreover, every open set in (E x E')/(G x G') is the image,
under the canonical mapping p : E x E' - (E x E')/(G x G'), .of an open set
U in E x E', and we have co(p(U)) = T^prj U) x 7i'(pr2 U), where n : E -> E/G
and n' : E' -* E'/G' are the canonical mappings. The set (o(p(U)) is therefore
open in (E/G) x (E'/G'), and the proof is complete.

(12.10.12)   Let G be a connected topological group acting continuously on a
topological space E. If the orbit space E/G is connected, then E is connected.

Since the mapping s\->s - x of G onto G  x is continuous, it follows
(3.19.7) that every orbit is connected. Suppose that there exist two non-empty
open sets U, V in E such that U n V = 0 and U u V = E. For each x e E,
the sets U n (G  x) and V n (G  x) are open in G  x; their union is G  jc
and their intersection is empty. Hence one of them is empty; in other words,
U and V are saturated. But this implies that 7i(U) and 7r(V) are nonempty
open sets in E/G whose intersection is empty and whose union is E/G, and
this contradicts the hypothesis that E/G is connected.

Remark

(12.10.1 3)   A right action of a group G on a set E is a mapping (s, x)\-*x- s
of G x E into E such that x - e = x for all x e E, and (x-t)-s = x' (ts) for allt subsets of E, whose union is E. For each z e n(An), let x be a point offt action) of G on E is a